Math

QuestionA right triangle has a hypotenuse of 5 inches. The longer leg is 1 inch more than the shorter leg. Find the lengths. Shorter leg: \square inches Longer leg: \square inches Hypotenuse: \square inches

Studdy Solution

STEP 1

Assumptions1. The hypotenuse of the right triangle is5 inches long. . The longer leg is1 inch longer than the shorter leg.
3. We are dealing with a right triangle, so we can use the Pythagorean theorem.

STEP 2

Let's denote the length of the shorter leg as xx. Therefore, the length of the longer leg would be x+1x +1.

STEP 3

We can set up the equation using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
(x+1)2+x2=52(x +1)^2 + x^2 =5^2

STEP 4

Now, we can simplify the equation.
x2+2x+1+x2=25x^2 +2x +1 + x^2 =25

STEP 5

Combine like terms.
2x2+2x+1=252x^2 +2x +1 =25

STEP 6

Subtract25 from both sides to set the equation to zero.
2x2+2x+125=02x^2 +2x +1 -25 =0

STEP 7

implify the equation.
2x2+2x24=02x^2 +2x -24 =0

STEP 8

We can divide the whole equation by2 to simplify it further.
x2+x12=0x^2 + x -12 =0

STEP 9

Now, we can factor the quadratic equation.
(x3)(x+4)=(x -3)(x +4) =

STEP 10

Setting each factor equal to zero gives the solutions to the equation.
x3=0orx+4=0x -3 =0 \quad or \quad x +4 =0

STEP 11

olving for xx gives x=3x =3 or x=4x = -4.

STEP 12

Since the length of a side cannot be negative, we discard x=4x = -4. Therefore, the length of the shorter side is inches.

STEP 13

Substitute x=3x =3 into x+x + to find the length of the longer side.
x+=3+=x + =3 + =So, the length of the longer side is inches.
Length of the shorter leg3 inchesLength of the longer leg inchesLength of the hypotenuse5 inches

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